Theorem cotr3 14944

Description: Two ways of saying a relation is transitive. (Contributed by RP, 22-Mar-2020.)

Hypotheses

Ref	Expression
cotr3.a	⊢ 𝐴 = dom 𝑅
cotr3.b	⊢ 𝐵 = (𝐴 ∩ 𝐶)
cotr3.c	⊢ 𝐶 = ran 𝑅

Assertion

Ref	Expression
cotr3	⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))

Distinct variable groups:   𝑥,𝑦,𝑧,𝑅   𝑥,𝐴,𝑦,𝑧   𝑦,𝐵,𝑧   𝑧,𝐶

Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥,𝑦)

Proof of Theorem cotr3

Step	Hyp	Ref	Expression
1		cotr3.a	. . 3 ⊢ 𝐴 = dom 𝑅
2	1	eqimss2i 4008	. 2 ⊢ dom 𝑅 ⊆ 𝐴
3		cotr3.b	. . . 4 ⊢ 𝐵 = (𝐴 ∩ 𝐶)
4		cotr3.c	. . . . 5 ⊢ 𝐶 = ran 𝑅
5	1, 4	ineq12i 4181	. . . 4 ⊢ (𝐴 ∩ 𝐶) = (dom 𝑅 ∩ ran 𝑅)
6	3, 5	eqtri 2752	. . 3 ⊢ 𝐵 = (dom 𝑅 ∩ ran 𝑅)
7	6	eqimss2i 4008	. 2 ⊢ (dom 𝑅 ∩ ran 𝑅) ⊆ 𝐵
8	4	eqimss2i 4008	. 2 ⊢ ran 𝑅 ⊆ 𝐶
9	2, 7, 8	cotr2 14943	1 ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∀wral 3044   ∩ cin 3913   ⊆ wss 3914   class class class wbr 5107  dom cdm 5638  ran crn 5639   ∘ ccom 5642

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649

This theorem is referenced by: (None)